Step-by-step Solution:

The slope of the line \(2x - 3y = 7\) is \[ \frac{2}{3} \] The required line is parallel to this line, so its slope will also be \(\frac{2}{3}\). The midpoint of the line segment joining the points \((1, 3)\) and \((1, -7)\) is \[ \left(\frac{1+1}{2}, \frac{3+(-7)}{2}\right) = (1, -2) \] The equation of the required line passing through \((1, -2)\) is \[ y + 2 = \frac{2}{3}(x - 1) \] Simplifying, \[ 3(y + 2) = 2(x - 1) \Rightarrow 3y + 6 = 2x - 2 \] \[ \Rightarrow 2x - 3y - 8 = 0 \]